In the thermodynamic limit, the number density can then be deduced from the partition function, Eq. (REF ) [1]}, [2]},
\(\textit {n} &=& \pm \sum _{i}\, \frac{g_i}{2 \pi ^2} \int _0^\infty \frac{(1-c)^{-1}\, (\gamma _q^{n_q})_i\;(\gamma _s^{n_s})_i\;\varepsilon _{c,d,r}(x_i) \; W_0\left[B(1-\frac{x_i}{r})^{\frac{1}{d}}\right]}{\left[1\pm (\gamma _q^{n_q})_i\;(\gamma _s^{n_s})_i\;\varepsilon _{c,d,r}(x_i)\right] \left(r-x_i\right) \left(1+W_0\left[B(1-\frac{x_i}{r})^{\frac{1}{d}}\right]\right)}\; \textbf {p}^2 d\textbf {p}.\) 
